AI 中文总结
该研究针对一维狄利克雷热方程，提出应用驱动的基准测试，比较11个内核，涵盖多种量子算法。通过不同设置分离误差，得出各算法在不同条件下的表现，如精度、误差等，为选择量子偏微分方程求解器提供实用指南。
AI 中文摘要
量子偏微分方程求解器在实践中难以评估，因为已发表的研究使用不同的离散化、输出模型、重建规则和硬件假设。我们针对一维狄利克雷热方程提出了一个可重现的、应用驱动的基准，在相同问题实例和读出协议下比较了11个内核。该基准涵盖相干线性求解器、变分量子线性求解器、虚时方法、实时哈密顿量模拟和酉扩张以及谱量子模拟方法。我们使用三种初始条件、四种网格大小、类似CFL的比率和最终时间。通过不同后端分离算法、采样和设备噪声误差。在状态向量上，谱量子模拟方法和施德 - 哈密顿量能将半离散参考精确到浮点精度，薛定谔化误差约为\(10^{-4}\)，虚时演化算法是处理平滑数据最强的非变换方法。在固定测量次数设置下，哈密顿量 - 哈罗 - 劳埃德算法相对\(\ell_2\)误差降至约\(0.79\)，而几种低深度或后选择方法受读出限制。范数不匹配消融将哈密顿量模拟、AVQDS和QLS - 傅里叶在\(n =7\)平滑初始条件下误差的23 - 29%归因于重建归一化。紧凑可观测量所需测量次数比全场重建少1 - 3个数量级。所得公共基准为选择量子偏微分方程求解器提供了实用指南。
英文摘要
Quantum PDE solvers are difficult to evaluate in practice because published studies use different discretizations, output models, reconstruction rules, and hardware assumptions. We present a reproducible, application-driven benchmark for the 1-D Dirichlet heat equation that compares eleven kernels under the same problem instances and readout contract. The benchmark covers coherent linear solvers (HHL, QSVT, and QLS-Fourier), VQLS, imaginary-time methods (QITE, var-QITE, and AVQDS), real-time Hamiltonian simulation and unitary dilations (Hamiltonian simulation, Schade-Hamiltonian, and Schr"odingerisation), and the spectral quantum simulation method (QSM). We use three initial conditions, four grid sizes from $n=4$ to $7$ qubits ($N=16$ to $128$), a CFL-like ratio $r\approx0.4$, and final time $T=1$. Statevector, ideal-shot ($10^5$ shots per step), and noisy Aer backends separate algorithmic, sampling, and device-noise errors. On statevector, QSM and Schade-Hamiltonian reproduce the semi-discrete reference to floating-point precision, Schr"odingerisation reaches approximately $10^{-4}$ error, and QITE is the strongest non-transform method for smooth data. Under the fixed-shot setting, HHL degrades to approximately $0.79$ relative $\ell_2$ error, while several low-depth or postselected methods become readout-limited. A norm-mismatch ablation attributes 23--29% of the $n=7$ smooth-initial-condition error of Hamiltonian simulation, AVQDS, and QLS-Fourier to reconstruction normalization. Compact observables, including total thermal energy and individual Fourier-mode weights, require 1--3 orders of magnitude fewer shots than full-field reconstruction. The resulting public benchmark provides a practical guide for selecting quantum PDE solvers.
Comments11 pages, 5 figures, 5 tables. Accepted for presentation at the 2026 IEEE International Conference on Quantum Computing and Engineering (QCE26, IEEE Quantum Week 2026); to appear in the conference proceedings
